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arXiv:2607.27470

Bose-Einstein condensation and superfluidity on a fuzzy sphere

Vira Shyta, Flavio S. Nogueira, Ashley M. Cook

Published Jul 29, 2026• 0 citations
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Abstract

According to Hohenberg's theorem, Bose-Einstein condensation (BEC) in two dimensions is impossible for any temperature $T>0$. By contrast, superfluidity does occur in two dimensions at finite temperatures; it emerges due to the breaking of Galilei invariance. Here we consider BEC and superfluidity on a compact two-dimensional space taking the form of a non-commutative ("fuzzy") sphere, where the scalar bosonic fields are promoted to $N\times N$ matrices. The dimension $N$ is related to the non-commutativity parameter of space and introduces an additional scale into the system. We find that non-commutativity favors ordered phases and so enhances BEC and superfluidity. We analyze BEC in ideal and weakly interacting Bose gases on a fuzzy sphere, finding in each case that the critical temperature of BEC is greater compared to that found in the case of a commutative sphere $S^2$. Then we investigate the superfluid response of weakly interacting Bose systems. To account for vortices in a superfluid, we show that, even on an ordinary sphere, the collective coordinates of vortices induce non-commutativity. With this in mind, we extend the definition of vortex defects to an inherently non-commutative sphere studied here, where the notion of a point is untenable. The non-commutativity is expected to be experimentally relevant to BEC and superfluidity since the fuzzy sphere has a thermodynamic limit distinct from the one defined over a plane, unlike the $S^2$ case. The significance of this difference is illustrated by the superfluid density calculation indicating that, in the large sphere limit, the normal fluid fraction on the fuzzy sphere yields a linear in $T$ dependence, while on a commutative $S^2$ it exhibits the usual two-dimensional $\sim T^3$ behavior. This linear dependence, arising directly from non-commutativity, is reminiscent of Uemura's law in cuprate high-$T_c$ superconductors.

DOI
10.48550/arxiv.2607.27470

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According to Hohenberg's theorem, Bose-Einstein condensation (BEC) in two dimensions is impossible for any temperature T>0T>0T>0. By contrast, superfluidity does occur in two dimensions at finite temperatures; it emerges due to the breaking of Galilei invariance. Here we consider BEC and superfluidity on a compact two-dimensional space taking the form of a non-commutative ("fuzzy") sphere, where the scalar bosonic fields are promoted to N×NN\times NN×N matrices. The dimension NNN is related to the non-commutativity parameter of space and introduces an additional scale into the system. We find that non-commutativity favors ordered phases and so enhances BEC and superfluidity. We analyze BEC in ideal and weakly interacting Bose gases on a fuzzy sphere, finding in each case that the critical temperature of BEC is greater compared to that found in the case of a commutative sphere S2S^2S2. Then we investigate the superfluid response of weakly interacting Bose systems. To account for vortices in a superfluid, we show that, even on an ordinary sphere, the collective coordinates of vortices induce non-commutativity. With this in mind, we extend the definition of vortex defects to an inherently non-commutative sphere studied here, where the notion of a point is untenable. The non-commutativity is expected to be experimentally relevant to BEC and superfluidity since the fuzzy sphere has a thermodynamic limit distinct from the one defined over a plane, unlike the S2S^2S2 case. The significance of this difference is illustrated by the superfluid density calculation indicating that, in the large sphere limit, the normal fluid fraction on the fuzzy sphere yields a linear in TTT dependence, while on a commutative S2S^2S2 it exhibits the usual two-dimensional ∼T3\sim T^3∼T3 behavior. This linear dependence, arising directly from non-commutativity, is reminiscent of Uemura's law in cuprate high-TcT_cTc​ superconductors.
Motivated by the recent achievement of space-based Bose-Einstein condensates (BEC) with ultracold alkali-metal atoms under microgravity and by the proposal of bubble traps which confine atoms on a thin shell, we investigate the BEC thermodynamics on the surface of a sphere. We determine analytically the critical temperature and the condensate fraction of a noninteracting Bose gas. Then we consider the inclusion of a zero-range interatomic potential, extending the noninteracting results at zero and finite temperature. Both in the noninteracting and interacting cases the crucial role of the finite radius of the sphere is emphasized, showing that in the limit of infinite radius one recovers the familiar two-dimensional results. We also investigate the Berezinski-Kosterlitz-Thouless transition driven by vortical configurations on the surface of the sphere, analyzing the interplay of condensation and superfluidity in this finite-size system.
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The Hohenberg theorem on the absence of Bose-Einstein condensation (BEC) in homogeneous systems of space dimensions D< 2D<~2D< 2 is based on a well-known Bogoliubov inequality. Applied to an assembly trapped in a harmonic potential we show that the Bogoliubov inequality rules out BEC in all dimensions at finite temperatures. However, this conflicting result with both theory and experiment disappears when the effect of the order parameter is properly taken into account in the boson-field commutation relations. For a hard-sphere Bose gas the theory is consistent with the expansion of the condensate when a positive scattering length is increased, as well as the collapse of the condensate when the sign of the scattering length is reversed and it reaches a minimum critical value.
We determine the critical temperature of hard-sphere Bosons in a cubic box by path-integral Monte Carlo simulations and finite-size scaling. We obtain an enhancement compared to the ideal gas at low densities in the form of a power law in density with exponent 1/31/31/3. At high densities the result for liquid helium, namely a lower critical temperature than in the non-interacting case, is recovered. We give a microscopic explanation for the observed behavior.
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The properties of phase transitions and the types of order present in the low-temperature states of matter are fundamentally dependent on the dimensionality of physical systems. Generally, highly ordered states are more robust in higher dimensions, while thermal and quantum fluctuations, which favour disordered states, play a more important role in lower dimensions. The case of a two-dimensional (2d) Bose fluid is particularly fascinating because of its "marginal" behaviour. In an infinite uniform 2d fluid thermal fluctuations at any non-zero temperature are strong enough to destroy the fully ordered state associated with Bose--Einstein condensation, but are not strong enough to suppress superfluidity in an interacting system at low, but non-zero temperatures. Further, the presence of residual "quasi-long-range" order at low temperatures leads to an interesting interplay between superfluidity and condensation in all experimentally relevant finite-size systems. In these notes we give an introduction to the physics of 2d Bose fluids from an atomic physics perspective. Our goal is to summarize the recent progress in theoretical understanding and experimental investigation of ultra-cold atomic gases confined to 2d geometry, and we also hope to provide a useful introduction to these systems for researchers working on related topics in other fields of physics.